Hippocampal neurons fire selectively in local behavioral contexts such as the position in an environment or phase of a task, and are thought to form a cognitive map of task-relevant variables. However, their activity varies over repeated behavioral conditions, such as different runs through the same position or repeated trials. Although widely observed across the brain, such variability is not well understood, and could reflect noise or structure, such as the encoding of additional cognitive information. Here, we introduce a conceptual model to explain variability in terms of underlying, population-level structure in single-trial neural activity. To test this model, we developed a novel unsupervised learning algorithm incorporating temporal dynamics, in order to characterize population activity as a trajectory on a nonlinear manifold--a space of possible network states. The manifold's structure captures correlations between neurons and temporal relationships between states, constraints arising from underlying network architecture and inputs. Using measurements of activity over time but no information about exogenous behavioral variables, we recovered hippocampal activity manifolds during spatial and non-spatial cognitive tasks in rats. Manifolds were low dimensional and smoothly encoded task-related variables, but contained an extra dimension reflecting information beyond the measured behavioral variables. Consistent with our model, neurons fired as a function of overall network state, and fluctuations in their activity across trials corresponded to variation in the underlying trajectory on the manifold. In particular, the extra dimension allowed the system to take different trajectories despite repeated behavioral conditions. Furthermore, the trajectory could temporarily decouple from current behavioral conditions and traverse neighboring manifold points corresponding to past, future, or nearby behavioral states. Our results suggest that trial-to-trial variability in the hippocampus is structured, and may reflect the operation of internal cognitive processes. The manifold structure of population activity is well-suited for organizing information to support memory, planning, and reinforcement learning. In general, our approach could find broader use in probing the organization and computational role of circuit dynamics in other brain regions.
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